You Are Designing A Study To Measure The Extent To Which Stu

You Are Designing A Study To Measure The Extent To Which Students In Y

You are designing a study to measure the extent to which students in your very large school district are planning to attend post-secondary school. You are trying to figure out how large a sample you need to be sure that your estimates are reasonably accurate (have modest-sized confidence intervals around the means). You remember from your evaluation course something about the relationship between the size of the sample and the size of that confidence interval. 23. If you double the sample size—say from 200 to 400 students—b. Give me a sense of how large the change in the width of the confidence interval around the impact estimate will be—for example, what is the order of magnitude of the change?

Paper For Above instruction

The relationship between sample size and the width of confidence intervals is fundamental in statistical research, especially in designing studies aimed at estimating population parameters such as proportions or means with reasonable accuracy. When assessing the planning of a study measuring students' intentions to attend post-secondary education, understanding how sample size influences the confidence interval's width helps determine the necessary sample size to achieve desired precision.

The mathematical relationship:

A confidence interval's width for an estimated proportion \( p \) (e.g., the proportion of students planning to attend post-secondary school) is approximately:

\[ \text{CI Width} \approx 2 \times Z_{\alpha/2} \times \sqrt{\frac{p(1 - p)}{n}} \]

where:

- \( Z_{\alpha/2} \) is the critical value from the standard normal distribution corresponding to the confidence level (e.g., 1.96 for 95% confidence),

- \( p \) is the estimated proportion,

- \( n \) is the sample size.

For simplicity, assuming \( p \) remains the same, the critical factor influencing the width of the confidence interval is \( \sqrt{1/n} \). This indicates that the width of the confidence interval is proportional to the inverse square root of the sample size.

Impact of doubling the sample size:

If initially, the sample size is \( n_1 = 200 \) and it is increased to \( n_2 = 400 \), the ratio of the confidence interval widths becomes:

\[ \frac{\text{Width at } n_2}{\text{Width at } n_1} = \frac{\sqrt{\frac{p(1 - p)}{400}}}{\sqrt{\frac{p(1 - p)}{200}}} = \frac{\sqrt{1/400}}{\sqrt{1/200}} = \frac{1/20}{1/\sqrt{200}} \]

Calculating:

\[ \frac{1/20}{1/\sqrt{200}} = \frac{1/20}{1/14.14} \approx \frac{14.14}{20} = 0.707 \]

Alternatively, directly considering the ratio:

\[ \frac{\sqrt{200}}{\sqrt{400}} = \frac{\sqrt{200}}{20} \approx \frac{14.14}{20} = 0.707 \]

Thus, doubling the sample size from 200 to 400 results in approximately a 29.3% reduction in the width of the confidence interval:

\[ 1 - 0.707 \approx 0.293 \text{ or } 29.3\% \]

Order of magnitude:

The reduction in confidence interval width follows an approximate inverse square root law with respect to sample size. This means that increasing the sample size by a factor of 4 would halve the confidence interval's width, whereas doubling it reduces the width to roughly 70% of its original size.

Implications for study design:

This relationship underscores that substantial increases in sample size are required to achieve modest improvements in estimate precision. For example, to cut the confidence interval width by half, the sample size must be increased by a factor of four. Conversely, doubling the sample size produces a roughly 30% reduction, emphasizing diminishing returns but still noticeable improvements.

Conclusion:

In practical terms, when planning to increase the sample size from 200 to 400, researchers can expect a reduction of about 29-30% in the confidence interval width around their estimate. This understanding allows for informed decisions balancing resource allocation against the desired accuracy of the findings.

References

  • Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.
  • Cozby, P. C., & Bates, S. C. (2015). Methods in Behavioral Research (12th ed.). McGraw-Hill Education.
  • Friedman, L. M., Furberg, C., & DeMets, D. L. (2010). Fundamentals of Clinical Trials. Springer.
  • Kish, L. (1965). Survey Sampling. John Wiley & Sons.
  • Lohr, S. L. (2009). Sampling: Design and Analysis. Cengage Learning.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (7th ed.). W.H. Freeman.
  • Proschan, M., & Wailand, P. (2014). Statistical Methods in Clinical Trials. Arnold.
  • Scheaffer, R. L., Mendenhall, W., Ott, L., & Gerow, K. G. (2012). Elementary Survey Sampling. Cengage Learning.
  • Sullivan, L. M. (2012). Essentials of Biostatistics in Public Health. Jones & Bartlett Learning.
  • Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson Education.

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